Rational solutions of an extended (2+1)-dimensional Camassa-Holm- Kadomtsev-Petviashvili equation in liquid drop

1.   Introduction

A classical dynamical system consists of a phase space $M$ together with a unique function $g$, where, by iterating this function, we obtain the orbits of points. However, we can find many systems with some finite maps rather than a single map that acts on the phase space. Indeed, we can find many natural processes involved with two or more interactions whose evolutions evolve with discrete time ^[1,2]. Therefore, there is a need to extend the study of dynamical systems by considering more than one mapping. Mathematicians have studied such systems either as non-autonomous systems or as iterated function systems (IFS). Therefore, these systems originate from a common study, specifically, the study of classical dynamical systems. Hence, important concepts in dynamics, including transitivity and shadowing ^[3], could be extended to IFSs.

In a dynamical system, generally, the future state follows from the initial state. Therefore, it is often deterministic. However, they often appear chaotic, i.e., minor changes in the initial state bring dramatically different long-term behavior. Both topological transitivity and shadowing are dynamical properties that are closely related to the chaoticity of dynamical systems. Usually, in chaos, topological transitivity is a part of its definition, or it is implied by it (at least in some spaces), or it implies chaos. Indeed, it is a part of the definition in Devaney's chaos ^[4], while in Li–Yorke chaos ^[5], if a function is topologically transitive (TT), then it is chaotic, but the converse is not valid. Moreover, a TT map $g$ has points that eventually move under iteration from one arbitrary small neighborhood to any other. As a result, one cannot break the corresponding system into a pair of invariant subsystems under $g$. Recently, mathematicians have studied this property intensively since it is a global characteristic in the dynamical systems theory.

Topological transitivity was introduced to dynamical systems by Birkhoff ^[6] in the 1920s. The term 'topologically transitive (TT)' is not a unified one. Instead, some authors use 'regionally transitive' ^[7,8], 'nomadic' ^[9], 'topologically ergodic' ^[10], cf. ^[8], and 'topologically indecomposable (or irreducible)' ^[11]. A mapping $g$ of a dynamical system $(M, g)$ is TT if, for any pair of non-empty open sets $W_1, W_2\subset M$, there is some $k > 0$ such that $g^k(W_1)\cap W_2\neq \emptyset$. The notion of topological transitivity was introduced to IFSs by Bahabadi in ^[3]. Devi and Mangang ^[12] have also discussed this notion in IFSs by giving several examples, and they also extend the notions of equicontinuity, sensitivity, and distality to IFSs. Moreover, Mangang ^[13] has studied the notions of mean equicontinuity, mean sensitivity, and mean distality of the product dynamical systems.

A natural generalization of topological transitivity is chain transitivity. It connects any two points of the phase space by a chain with any desired error bound. It is an essential notion of a dynamical system. For example, if a dynamical system is chain transitive, then several shadowing properties, including thick shadowing and shadowing, are equivalent ^[14]. In a dynamical system, there might be a circumstance where for any error bound $\gamma$, we could not find a $\gamma$-chain but it may be simpler to obtain an $\eta$-average (or $\eta$-mean) chain with any average (or mean) error bound $\gamma$. It leads to the introduction of average (or mean) chain transitivity to dynamical systems.

In dynamical systems theory, our main goal is to study the nature of all its orbits. Likewise, in IFSs, we study the orbit behaviors of the system. Yet, in particular cases, it is unlikely to compute the accurate initial value of a point, which gives rise to the approximate values of the orbits. Thus, we obtain pseudo-orbits of the system. The notion of shadowing puts these pseudo-orbits close to the actual orbits of the system. It was introduced independently by Anosov ^[15] and Bowen ^[16] in the 1970s. Shadowing plays an essential part in developing the qualitative theory of dynamical systems. In systems with shadowing property, any pseudo-orbit is followed uniformly by a true orbit over an arbitrarily long duration of time. Usually, it is crucial in systems with chaos, where even an arbitrarily small error in the initial position leads to a large divergence of orbits. Moreover, the shadowing lemma in ^[16] roughly states that shadowing is a common phenomenon in chaotic dynamical systems. In recent years, shadowing has developed intensively and has become a notion of great interest. Many researchers have introduced different aspects of shadowing in dynamical systems, including average shadowing ^[17], $h$-shadowing ^[18], ergodic shadowing ^[19], thick shadowing ^[20], and d-shadowing ^[20]. Consequently, these aspects of shadowing have also been extended to IFSs; for references, one can see, ^[3,21,22,23].

Ruchi Das and Mukta Garg introduced the notions of average (or mean) chain properties and the almost average (or mean) shadowing property to dynamical systems in ^[24]. Unlike the classical shadowing property, the notion of the almost mean shadowing property deals with pseudo-orbits with very small mean errors. In ^[25], the authors have also investigated the chaotic behavior of maps with almost average (or mean) shadowing property.

Motivated by this, in this work we wish to study the concepts of mean chain properties and the almost mean shadowing property in IFSs. In Section 2, we give some preliminary discussions on dynamical systems and IFSs. In Section 3, we introduce the notions of mean chain transitive (MCT), mean chain mixing (MCM), and totally mean chain transitive (TMCT) to IFSs and study the relations among them. We also give an example of an IFS that is not chain transitive (CT) but MCM (Example 3.5). In Section 4, we introduce the notion of almost mean shadowing property (AMSP) to IFSs and study some of its basic properties. We also study the relation between CT and AMSP in IFSs. In particular, in Theorem 4.7, we find that an IFS is CT if one of the constituent maps is surjective, and it has AMSP.

2.   Preliminaries

We consider $(M, g)$ to be a dynamical system where, $(M, d)$ is a compact metric space and $f$ is a self-continuous map on $M$. Put $\mathbb{Z}_+ = \{n\in\mathbb{Z}:n\geq0\}\}$. Then the set $O(s, g) = \{g^{n}(s):n\in\mathbb{Z}_+\}$ is said to be the orbit of $s\in M$ under $(M, g)$.

Let $(M, g)$ be a dynamical system. Let $\gamma, \eta > 0$, then

ⅰ) A finite sequence $\{s_0, s_1, \ldots, s_n\}$ in $M$ is an $\eta$-chain if $d(g(s_i), s_{i+1}) < \eta$, $\forall$ $0\leq i\leq n-1$. When $i$ is not bounded above, it is called an $\eta$-pseudo-orbit.

ⅱ) An $\eta$-pseudo-orbit $\{s_i\}_{i\in\mathbb{Z}_+}$ is $\gamma$-shadowed by $s\in M$ if $d(g^i(s), s_i) < \gamma$, $\forall$ $i\geq0$.

A dynamical system $(M, g)$ is said

a) To have shadowing property (SP) if $\forall$ $\gamma > 0$, $\exists$ $\eta > 0$ such that every $\eta$-pseudo-orbit is $\gamma$-shadowed by some point in $M$.

b) To be chain transitive (CT) if $\forall$ $\eta > 0$, and for any pair of points $s, t\in M$, $\exists$ an $\eta$-chain joining $s$ and $t$.

c) To be chain mixing (CM) if $\forall$ $\eta > 0$, and for any pair of points $s, t\in M$, $\exists$ $N > 0$, such that $\forall$ $n\geq N$, $\exists$ an $\eta$-chain joining $s$ and $t$ of length $n$.

Hutchinson introduced IFSs in ^[26] and were popularized by Barnsley ^[27]. Moreover, Barnsley and Demko ^[28] first named the word IFS, and it has garnered much attention since then. Let $\Lambda$ be a non-empty finite set; an IFS $\mathfrak{F} = \{M; g_{\alpha}|\alpha\in\Lambda\}$ is a family of continuous mappings $g_{\alpha}:M\rightarrow M$, where $\alpha\in\Lambda$, and $(M, d)$ is a compact metric space. Put $\Lambda^{{\mathbb{Z}}_+} = \{\langle \alpha_i \rangle:\alpha_i\in\Lambda\; \forall i\in\mathbb{Z}_+\}$. We use the short notation

Let $\sigma = \langle \alpha_i \rangle$ be a typical member of $\Lambda^{{\mathbb{Z}}_+}$. An infinite sequence $\{s_i\}_{i\in{\mathbb{Z}_+}}$ in $M$ is an orbit of $\mathfrak{F}$ if $\exists$ $\sigma\in\Lambda^{{\mathbb{Z}}_+}$, such that $s_i = \mathfrak{F}_{\sigma_i}(s_0)$, where $\mathfrak{F}_{\sigma_i}(s_0) = g_{\alpha_{i-1}}\circ g_{\alpha_{i-2}}\circ\cdots\circ g_{\alpha_1}\circ g_{\alpha_0}(s_0)$ and $\mathfrak{F}_{\sigma_0}(s_0) = s_0$. So, for any $\sigma\in\Lambda^{\mathbb{Z}_+}$, we define $O_\sigma(s) = \{\mathfrak{F}_{\sigma_i}(s):i\in\mathbb{Z}_+\}$ as an orbit of $s\in M$ related to $\sigma$. For an IFS $\mathfrak{F}$ and for a fixed integer $n > 0$, we define $\Lambda^n = \{(\alpha_0, \alpha_1, \ldots, \alpha_{n-1}):\alpha_i\in\Lambda, \; 0\leq i\leq n-1\}$; $f_\mu = g_{\alpha_{n-1}}\circ\cdots\circ g_{\alpha_1}\circ g_{\alpha_0}$; and $\mathfrak{F}^n = \{f_\mu\; |\; \mu\in\Lambda^n\}$.

Bahabadi ^[3] extended the notions of SP, average (or mean) SP (MSP), TT, CT and CM to IFSs. An IFS $\mathfrak{F}$ is TT if for any pair of non-empty open sets $W_1, W_2 \subset M$, $\exists$ $\sigma\in\Lambda^{\mathbb{Z}_+}$ such that $\mathfrak{F}_{\sigma_k}(W_1)\cap W_2 \neq \emptyset$ for some $k\geq0$.

Let $\mathfrak{F} = \{M; g_{\alpha}|\alpha\in\Lambda\}$ be an IFS and let $\gamma, \eta > 0$, then

ⅰ) A finite sequence $\{s_0, s_1, \ldots, s_n\}$ in $M$ is an $\eta$-chain if $\exists$ $\{\alpha_0, \alpha_1, \ldots, \alpha_{n-1}\}$ such that $d(g_{\alpha_i}(s_i), s_{i+1}) < \eta$, $\forall$ $0\leq i\leq n-1$. When $i$ is not bounded above, it is called an $\eta$-pseudo-orbit.

ⅱ) An $\eta$-pseudo-orbit $\{s_i\}_{i\in\mathbb{Z}_+}$ is $\gamma$-shadowed by $s\in M$ if $\exists$ $\sigma\in\Lambda^{\mathbb{Z}_+}$ such that $d(\mathfrak{F}_{\sigma_i}(s), s_i) < \gamma$, $\forall$ $i\geq0$.

ⅲ) $\{s_i\}_{i\in\mathbb{Z}_+}$ is an $\eta$-mean-pseudo-orbit if $\exists$ $N > 0$, and $\exists$ $\sigma\in\Lambda^{\mathbb{Z}_+}$ such that for any $n\geq N$,

ⅳ) An $\eta$-mean-pseudo-orbit $\{s_i\}_{i\in\mathbb{Z}_+}$ is $\gamma$-mean shadowed by $s\in M$ if

An IFS $\mathfrak{F} = \{M; g_{\alpha}|\alpha\in\Lambda\}$ is said

a) To have SP if $\forall$ $\gamma > 0$, $\exists$ $\eta > 0$ such that every $\eta$-pseudo-orbit is $\gamma$-shadowed by a point in $M$.

b) To be CT if $\forall$ $\eta > 0$, and for any pair of points $s, t\in M$, $\exists$ an $\eta$-chain joining $s$ and $t$.

c) To be CM if $\forall$ $\eta > 0$ and for any pair of points $s, t\in M$, $\exists$ $N > 0$ such that $\forall$ $n\geq N$, $\exists$ an $\eta$-chain joining $s$ and $t$ of length $n$.

d) To have mean shadowing property (MSP) if $\forall$ $\gamma > 0$, $\exists$ $\eta > 0$ such that every $\eta$-mean-pseudo-orbit is $\gamma$-mean shadowed by a point in $M$.

Mean chain properties and almost mean shadowing properties in dynamical systems have been introduced in ^[24]. The main aim of this paper is to extend these notions in IFSs. Therefore, we recall the following definitions in dynamical systems:

Let $\eta > 0$, a finite sequence $\{s_0, s_1, \ldots, s_n\}$ is an $\eta$-mean chain of length $n$ if $\exists$ an integer $0 < P\leq n$ such that $\forall$ $P\leq m\leq n$,

$(M, g)$ is said to be mean chain transitive (MCT) if, for every $\eta > 0$ and for any pair of points $s, t\in M$, $\exists$ an $\eta$-mean chain joining $s$ and $t$. It is said to be totally mean chain transitive (TMCT) if $g^k$ is MCT for each $k > 0$. And, it is said to be mean chain mixing (MCM) if for every $\eta > 0$ and for any pair of points $s, t\in M$, $\exists$ an integer $N > 0$ such that $\forall$ $n\geq N$, $\exists$ an $\eta$-mean chain joining $s$ and $t$ of length $n$.

Let $\eta > 0$, a sequence $\{s_i\}_{i\in\mathbb{Z}_+}$ is an almost $\eta$-mean pseudo-orbit of $(M, g)$ if

An almost $\eta$-mean pseudo-orbit $\{s_i\}_{i\in\mathbb{Z}_+}$ is $\gamma$-mean shadowed by $s\in M$ if

$(M, g)$ has almost mean shadowing property (AMSP) if for every $\gamma > 0$, $\exists$ an $\eta > 0$ such that every almost $\eta$-mean pseudo-orbit $\{s_i\}_{i\in\mathbb{Z}_+}$ is $\gamma$-mean shadowed by a point in $M$. Throughout the paper, we consider $(M, d)$ to be a compact metric space and $g_\alpha:X\rightarrow X$ to be a continuous self-map in $X$ for any $\alpha\in\Lambda$.

3.   Mean chain properties

This section introduces the notions of mean chain transitivity (MCT), mean chain mixing (MCM), and totally mean chain transitivity (TMCT) to IFSs, and proves some preliminary results.

Definition 3.1. Let $\eta > 0$, a finite sequence $\{s_0, s_1, \ldots, s_n\}$ is an $\eta$-mean chain of length $n$ if $\exists$ $\{\alpha_0, \alpha_1, \ldots, \alpha_{n-1}\}$ and an integer $0 < P\leq n$ such that $\forall$ integer $m$ with $P\leq m\leq n$,

Definition 3.2. An IFS $\mathfrak{F}$ is considered to be MCT if for any $\eta > 0$ and for any pair of points $s, t\in M$, $\exists$ an $\eta$-mean chain joining $s$ and $t$.

Definition 3.3. An IFS $\mathfrak{F}$ is considered to be TMCT if $\mathfrak{F}^k$ is MCT for each $k > 0$.

Definition 3.4. An IFS $\mathfrak{F}$ is considered to be MCM if for any $\eta > 0$ and for any pair of points $s, t\in M$, $\exists$ an integer $N > 0$ such that $\forall$ $n\geq N$, $\exists$ an $\eta$-mean joining $s$ and $t$ of length $n$.

Following, we give an example of an IFS that is not CT but MCM.

Example 3.5. Consider $(M, d)$ to be a metric space with more than two elements, and let $a, b\in M$. Let $g_1, g_2$ be two self-maps in $M$ defined by $g_1(s) = a$ and $g_2(s) = b$ for every $s\in M$. Then, the IFS, $\mathfrak{F} = \{M; g_1, g_2\}$ is not CT but MCM.

Proof. Clearly, $\mathfrak{F}$ is not CT, indeed for any pair $s, t\in M$ with $t\notin\{a, b\}$, there is no $\eta$-chain joining $s$ and $t$ with $\eta < \min\{d(t, a), d(t, b)\}$. Now, we claim that $\mathfrak{F}$ is MCM. Take $\eta > 0$ and $s, t\in M$. For $t\in\{a, b\}$, it is obvious. Suppose $t\notin\{a, b\}$ and, let $\max\{d(t, a), d(t, b)\} = \gamma > 0$. Choose an integer $N > 0$ for which $N > \frac{\gamma}{\eta}$. For every $n\geq N$ and a finite sequence $\{\alpha_i\}_{i = 0}^{n-1}$ where $\alpha_i\in\{1, 2\}$, define $s_i = \mathfrak{F}_{\sigma_i}(s)$ for $0\leq i\leq n-1$ and $s_n = t$. Then, for every integer $m$ with $N\leq m\leq n$, we have

Thus, $\{s_i\}_{i = 0}^n$ is an $\eta$-mean chain joining $s$ and $t$ of length $n$. Hence, $\mathfrak{F} = \{M; g_1, g_2\}$ is not CT but MCM. □

Theorem 3.6. Let $\mathfrak{F}$ be an IFS. Then, $\mathfrak{F}$ is MCT if $\mathfrak{F}^k$ is MCT for some $k > 1$.

Proof. Let $\eta > 0$ and let $s, t\in M$ be two points. Let $k > 1$ be fixed such that $\mathfrak{F}^k$ is MCT. Then, there exists an $\eta$-mean chain $\{s_i\}_{i = 0}^n$ of $\mathfrak{F}^k$ joining $s$ and $t$. Therefore, there exists a finite sequence $\{\mu_0, \mu_1, \ldots, \mu_{n-1}\}$ and an integer $0 < P\leq n$ such that $\forall$ integer $m$ with $P\leq m\leq n$,

where $f_{\mu_i} = g_{\alpha_{k-1}^i}\circ g_{\alpha_{k-2}^i}\circ\cdots\circ g_{\alpha_{0}^i}$ and $\mu_i = \{\alpha_{0}^i, \alpha_{1}^i, \ldots, \alpha_{k-1}^i\}$ for $0\leq i\leq n-1$.

For $0\leq i\leq n$, let

i.e.,

Again, let

Then, $\forall$ integer $l$ with $n\leq l\leq nk$, we have

For $j\neq ik$, the term vanishes, therefore

By using (3.1), we have

Thus, $\{t_j\}_{j = 0}^{nk}$ is an $\eta$-mean chain joining $s$ and $t$ of length $nk$. Hence $\mathfrak{F}$ is MCT. □

Theorem 3.7 shows that a TMCT IFS is MCM if the constituent maps are Lipschitz. A self-continuous function $g$ on a metric space $M$ is a Lipschitz function if $\exists$ $L > 0$ such that $d(g(s), g(t))\leq Ld(s, t)$, $\forall$ $s, t\in M$.

Theorem 3.7. Let $\mathfrak{F} = \{M; g_\alpha|\alpha\in\Lambda\}$ be an IFS where each $g_\alpha$ is a Lipschitz function. If $\mathfrak{F}$ is MCM, then $\mathfrak{F}$ is TMCT.

Proof. Let $k > 1$ be an integer, let $\eta > 0$ and let $s, t\in M$ be any pair of points. For each $\alpha\in\Lambda$, as $g_\alpha$ is Lipschitz, $\exists$ $L_\alpha > 0$ such that $d(g_\alpha(u), g_\alpha(v))\leq L_\alpha d(u, v)$, $\forall$ $u, v\in M$. Let $L =$max$\{L_\alpha:\alpha\in\Lambda\}$. Then, $d(g_\alpha(u), g_\alpha(v))\leq L d(u, v)$, $\forall$ $\alpha\in\Lambda$ and $\forall$ $u, v\in M$. Without loss of generality, let $L\geq1$ and take $\gamma = \frac{\eta}{kL^{k-1}}$. Since $\mathfrak{F}$ is MCM, $\exists$ $N > 0$ such that $\forall$ $n\geq N$, $\exists$ a $\gamma$-mean chain of $\mathfrak{F}$ joining $s$ and $t$ of length $n$. Take an integer $r > 0$ such that $rk\geq N$. Then, we can get a $\gamma$-mean chain of $\mathfrak{F}$ joining $s$ and $t$ of length $rk$, say $\{s_0 = s, s_1, \ldots, s_{rk} = t\}$. Therefore, there exists an integer $0 < P\leq rk$ and a finite sequence, say $\{\alpha_i\}_{i = 0}^{rk-1} = \{\alpha_0^0, \alpha_1^0, \ldots, \alpha_{k-2}^0, \alpha_{k-1}^0, \alpha_0^1, \alpha_1^1, \ldots, \alpha_{k-2}^1, \alpha_{k-1}^1, \ldots, \alpha_{k-2}^{r-1}, \alpha_{k-1}^{r-1}\}$ such that $\forall$ integer $m$ with $P\leq m\leq rk$,

Put $\gamma_i = d(g_{\alpha_i(s_i)}, s_{i+1})$ for $0\leq i\leq rk-1$. Then, from Eq (3.2), we get

Define $t_i = s_{ik}$ for $0\leq i\leq r$. We claim that $\{t_0, t_1, \ldots, t_r\}$ is an $\eta$-mean chain of $\mathfrak{F}^k$ joining $s$ and $t$.

Let $f_{\mu_i} = g_{\alpha_{k-1}^i}\circ g_{\alpha_{k-2}^i}\circ\cdots\circ g_{\alpha_{0}^i}$ where $\mu_i = \{\alpha_{0}^i, \alpha_{1}^i, \ldots, \alpha_{k-1}^i\}$, $\forall$ $0\leq i\leq r-1$. Then, $\forall$ $0\leq i\leq r-1$, we have

Therefore, using Eq (3.3), it is clear that

Thus, $\{t_0, t_1, \ldots, t_r\}$ is an $\eta$-mean chain of $\mathfrak{F}^k$ joining $s$ and $t$. Hence, $\mathfrak{F}$ is TMCT. □

Given two compact metric spaces $(M, d)$ and $(M', d^{'})$, we take the metric space $M\times M'$ with metric

and let $\mathfrak{F} = \{M, g_\alpha|\alpha\in\Lambda\}$ and $\mathfrak{G} = \{M'; g_\beta|\beta\in\Gamma\}$ be two IFSs.

Then, we define the IFS, $\mathfrak{F}\times\mathfrak{G}$ as

where $h_{\alpha, \beta}(s, t) = (g_\alpha(s), g_\beta(t))$, $\forall$ $s\in M$ and $t\in M'$.

Theorem 3.8. If $\mathfrak{F} = \{M; g_\alpha|\alpha\in\Lambda\}$ is a MCM IFS, then $\mathfrak{F}\times\mathfrak{F}$ is MCT.

Proof. Let $\eta > 0$ and let $(s, t), (u, v)\in M\times M$ be any two points.

Since $\mathfrak{F}$ is MCM and as $\frac{\eta}{2} > 0$, there exist integers $N_1, \; N_2 > 0$ such that for any $n_1\geq N_1$ and $n_2\geq N_2$, there are $\frac{\eta}{2}$-mean chains joining $s$ and $u$, and joining $t$ and $v$ respectively.

Put $N =$max$\{N_1, N_2\}$. Then, we can find two $\frac{\eta}{2}$-mean chains joining $s$ and $u$, $t$ and $v$ respectively; say $\{s_0 = s, s_1, \ldots, s_N = u\}$ and $\{t_0 = t, t_1, \ldots, t_N = v\}$. Therefore, there exist finite sequences $\{\alpha_0, \alpha_1, \ldots, \alpha_{N-1}\}$ and $\{\alpha_0^{'}, \alpha_1^{'}, \ldots, \alpha_{N-1}^{'}\}$, and integers $0 < P_1, \; P_2\leq N$ such that $\forall$ integers $m', m''$ with $P_1\leq m'\leq N$ and $P_2\leq m''\leq N$, we have

and

Take $P =$max$\{P_1, \; P_2\}$ and consider $\{(s_i, t_i)\}_{i = 0}^N$. Then, $\forall$ integer $m$ with $P\leq m\leq N$, we have

Thus, $\{(s_i, t_i)\}_{i = 0}^N$ is an $\eta$-mean chain of $\mathfrak{F}\times\mathfrak{F}$ joining $(s, t)$ and $(u, v)$ of length $N$. Hence, $\mathfrak{F}\times\mathfrak{F}$ is MCT. □

Theorem 3.9. If $\mathfrak{F}$ is a TMCT IFS, then $\mathfrak{F}\times\mathfrak{F}$ is MCT.

Proof. Let $\eta > 0$ and let $(s, t), (u, v)\in M\times M$ be any two points. By definition of TMCT, we have $\mathfrak{F}$ is MCT. Suppose, $\{s_0 = s, s_1, \ldots, s_n = u\}$ and $\{u_0 = u, u_1, \ldots, u_k = u\}$ are two $\frac{\eta}{4}$-mean chains respectively, joining $s$ and $u$, and joining $u$ to itself. Then, there exist finite sequences $\{\alpha_0, \alpha_1, \ldots, \alpha_{n-1}\}$ and $\{\alpha_0^{'}, \alpha_1^{'}, \ldots, \alpha_{k-1}^{'}\}$ and integers $0 < P_1\leq n$ and $0 < P_2\leq k$ such that

and

In particular,

and

From the definition of TMCT, $\mathfrak{F}^k$ is MCT. Let $\{t_0 = g_{\alpha_{n-1}}\circ\cdots\circ g_{\alpha_0}(t), t_1, \ldots, t_p = v\}$ be an $\frac{\eta}{2}$-mean chain of $\mathfrak{F}^k$ joining $g_{\alpha_{n-1}}\circ\cdots\circ g_{\alpha_0}(t)$ and $v$ of length $p$. Therefore, we can find a finite sequence $\{\mu_0, \mu_1, \ldots, \mu_{p-1}\}$ and an integer $0 < P\leq p$ such that

where $f_{\mu_i} = g_{\alpha_{k-1}^i}\circ g_{\alpha_{k-2}^i}\circ\cdots\circ g_{\alpha_{0}^i}$ and $\mu_i = \{\alpha_{0}^i, \alpha_{1}^i, \ldots, \alpha_{k-1}^i\}$ for $0\leq i\leq p-1$.

Consider

with respect to the finite sequence

Then, it is clear that the term $d(g_{\alpha_i^{''}}(z_i), z_{i+1})$ vanishes whenever $i\neq n+jk$ where $0 < j\leq p-1$. Therefore,

Thus, $\{z_i\}_{i = 0}^{n+pk}$ is an $\frac{\eta}{2}$-mean chain of $\mathfrak{F}$ joining $t$ and $v$.

Again, consider

with respect to the finite sequence

Now,

Thus, $\{w_i\}_{i = 0}^{n+pk}$ is an $\frac{\eta}{2}$-mean chain of $\mathfrak{F}$ joining $s$ and $u$. This implies that $\{(w_i, z_i)\}_{i = 0}^{n+pk}$ is an $\frac{\eta}{2}$-mean chain of $\mathfrak{F}\times\mathfrak{F}$ joining $(s, t)$ and $(u, v)$ with respect to $d^{''}$. Hence, $\mathfrak{F}\times\mathfrak{F}$ is MCT. □

4.   Almost mean shadowing property

This section introduces the notion of almost mean shadowing property (AMSP) to IFSs.

Definition 4.1. Let $\eta > 0$, a sequence $\{s_i\}_{i\in\mathbb{Z}_+}$ is an almost $\eta$-mean pseudo-orbit of an IFS $\mathfrak{F}$ if $\exists$ $\sigma\in\Lambda^{\mathbb{Z}_+}$ such that

An almost $\eta$-mean pseudo-orbit $\{s_i\}_{i\in\mathbb{Z}_+}$ of an IFS $\mathfrak{F}$ is $\gamma$-mean shadowed by $s\in M$ if

Definition 4.2. An IFS $\mathfrak{F}$ has AMSP if for any $\gamma > 0$, $\exists$ an $\eta > 0$ such that every almost $\eta$-mean pseudo-orbit $\{s_i\}_{i\in\mathbb{Z}_+}$ is $\gamma$-mean shadowed by a point in $M$.

Remark 4.3. From the definition, it is clear that AMSP implies MSP, but the converse may not be true.

In the following, we give an example of an IFS that has MSP but does not have the AMSP.

Example 4.4 Let $(M, d)$ be the metric space as defined in [29, Example 9.1]. Let $g_1, \; g_2$ be self maps on $M$ defined as

Then, the IFS, $\mathfrak{F} = \{M; g_1, g_2\}$ has MSP but does not have the AMSP.

Proof. Proceeding similarly, as in the proof of [29, Theorem 9.2], it is clear that $\mathfrak{F}$ has MSP.

Also, in ^[24], it is given that $(M, g_1)$ does not have the AMSP. So, for any $\epsilon > 0$, we can find a $\delta > 0$ and an almost $\delta$-pseudo orbit with respect to $\sigma = \{g_1, g_1, g_1, \cdots\}$ which is not $\epsilon$-shadowed in average by any point in $M$. Hence, the IFS $\mathfrak{F} = \{M; g_1, g_2\}$ does not have the AMSP. □

Example 4.5. Let $\mathfrak{F}$ be the IFS as defined in [30, Example 3.5]. Then, $\mathfrak{F}$, does not have MSP and AMSP.

Proof. In [31, Remark 4.5], it has been given that $\mathfrak{F}$ does not have MSP. Using, the above Remark 4.3, it is clear that $\mathfrak{F}$ does not have the AMSP. □

Theorem 4.6. If $\mathfrak{F}$ is an IFS with AMSP, then so does $\mathfrak{F}^k$ for every $k\geq2$.

Proof. Let $k\geq2$ and $\gamma > 0$. By hypothesis, $\exists$ an $\eta > 0$ such that every almost $\eta$-mean pseudo-orbit is $\frac{\gamma}{k}$-mean shadowed by a point in $M$.

Let $\{t_i\}_{i\in\mathbb{Z}_+}$ be an almost $\eta$-mean pseudo-orbit of $\mathfrak{F}^k$. Then, $\exists$ $\sigma = \langle \mu_i\rangle$ such that $\forall$ $\mu_i\in\sigma$

where $f_{\mu_i} = g_{\alpha_{k-1}^i}\circ g_{\alpha_{k-2}^i}\circ\cdots\circ g_{\alpha_{0}^i}$ and $\mu_i = \{\alpha_0^i, \alpha_1^i, \ldots, \alpha_{k-1}^i\}\in\Lambda^k$, $\forall$ $i\in\mathbb{Z}_+$.

Now, for some $\sigma^{'} = \langle\alpha_j^{'}\rangle = \{\alpha_0^0, \alpha_1^0, \ldots, \alpha_{k-1}^0, \alpha_0^1, \ldots\}$, consider a sequence $\{s_j\}_{j\in\mathbb{Z}_+}$ defined by

For $ki < j < (i+1)k$, we have $0 < l\leq k-1$ such that $s_j = s_{ik+l} = g_{\alpha_{l-1}^i}\circ g_{\alpha_{l-2}^i}\circ\cdots\circ g_{\alpha_0^i}(t_i)$. Also, for any integer $n > 0$, we can get some $i\geq0$ and $0 < l\leq k-1$ for which $n = ik+l$. Thus,

For $j\neq ki-1$, the term vanishes. Therefore,

This implies that $\{s_j\}_{j\in\mathbb{Z}_+}$ is an almost $\eta$-mean pseudo-orbit of $\mathfrak{F}$ with respect to $\sigma^{'}$. Therefore, $\exists$ $z\in M$ such that

Now,

Hence, $\mathfrak{F}^k$ hasAMSP for every $k\geq2$. □

Theorem 4.7. Let $\mathfrak{F} = \{M; g_\alpha|\alpha\in\Lambda\}$ be an IFS, where one of the $g_\alpha$ is surjective. If $\mathfrak{F}$ has AMSP, then it is CT.

Proof. Let $\gamma > 0$ and let $s, t\in M$ be two points. Since $\{g_\alpha:\alpha\in\Lambda\}$ is a family of uniformly continuous mappings, it is uniformly equicontinuous. Thus, $\exists$ $0 < \eta < \gamma$ such that $\forall$ $u, v\in M$ and $\forall$ $\alpha\in\Lambda$, $d(g_\alpha(u), g_\alpha(v)) < \gamma$ whenever $d(u, v) < \eta$. By hypothesis, $\mathfrak{F}$ has AMSP. Therefore, $\exists$ $\delta > 0$ such that every almost $\delta$-mean pseudo-orbit of $\mathfrak{F}$ is $\frac{\eta}{2}$-mean shadowed by a point in $M$.

Let $D =$diam$(M)$ and let $K > 0$ be an integer such that $\frac{D}{K} < \delta$. Suppose for a fixed $\alpha^{*}\in\Lambda$, $g_{\alpha^{*}}$ is surjective. Then, we can easily see that $g_{\alpha^{*}}^{-l}(t)$ exists $\forall$ integer $l$ with $0\leq l\leq K-1$.

For $i\in \mathbb{Z}_+$, fix an infinite sequence $\sigma = \langle\alpha_i\rangle\in\Lambda^{\mathbb{Z}_+}$. Again, for $j\in\mathbb{Z}_+$, we consider an infinite sequence $\{s_j\}_{j\in\mathbb{Z}_+}$, where

For any $\alpha'\in\Lambda$, let us define $\sigma^{'} = \langle\alpha_j^{'}\rangle\in\Lambda^{\mathbb{Z}_+}$, where

Now, for any $n > 0$ with $iK\leq n\leq (i+1)K$, we have

Therefore,

This implies that $\{s_j\}_{j\in\mathbb{Z}_+}$ is an almost $\delta$-mean pseudo-orbit of $\mathfrak{F}$. By hypothesis, $\exists$ $z\in M$ such that

Notice that, there exist infinitely many $i\in\mathbb{Z}_+$ for which there is some $l$ with $2iK\leq l\leq (2i+1)K-1$, i.e., $s_{l}\in\{s, \mathfrak{F}_{\sigma_1}(s), \mathfrak{F}_{\sigma_2}(s), \ldots, \mathfrak{F}_{\sigma_{K-1}}(s)\}$ such that $d\left(\mathfrak{F}_{\sigma_{l}^{'}}(z), s_{l}\right) < \frac{\eta}{2}$. Otherwise,

which contradicts (4.1).

Similarly, the above statement holds when $(2i+1)K\leq l\leq2(i+1)K-1$, i.e., $s_{l}\in\{g_{\alpha^{*}}^{-(K-1)}(t), g_{\alpha^{*}}^{-(K-2)}(t), \ldots, g_{\alpha^{*}}^{-1}(t), t\}$.

Thus, we can find two integers $l_1$ and $l_2$ with $0 < l_1 < l_2$ such that $s_{l_1} = \mathfrak{F}_{\sigma_{p_1}}(s)$ for some $0\leq p_1\leq K-1$ satisfying $d\left(\mathfrak{F}_{\sigma_{l_1}^{'}}(z), s_{l_1}\right) < \frac{\eta}{2}$ and $s_{l_2} = g_{\alpha^{*}}^{-p_2}(t)$ for some $0\leq p_2\leq K-1$ satisfying $d\left(\mathfrak{F}_{\sigma_{l_2}^{'}}(z), s_{l_2}\right) < \frac{\eta}{2}$. Using the condition of equicontinuity, we have, $d\left(g_{\alpha_{l_1}^{'}}(\mathfrak{F}_{\sigma_{l_1}^{'}}(z)), g_{\alpha_{l_1}^{'}}(s_{l_1})\right) < \gamma$. This implies that $d\left(\mathfrak{F}_{\sigma_{l_1+1}^{'}}(z), g_{\alpha_{l_1}^{'}}(s_{l_1})\right) < \gamma$ and $d\left(\mathfrak{F}_{\sigma_{l_2}^{'}}(z), s_{l_2}\right) < \gamma$. Therefore,

$\mathfrak{F}_{\sigma_{l_1+2}^{'}}(z), \ldots, \mathfrak{F}_{\sigma_{l_2-1}^{'}}(z), g_{\alpha^{*}}^{-p_2}(t) = s_{l_2}, g_{\alpha^{*}}^{-(p_2-1)}(t), \ldots, g_{\alpha^{*}}^{-1}(t), t\}$ is a $\gamma$-chain joining $s$ and $t$ with respect to the finite sequence $\{\alpha_0, \alpha_1, \ldots, \alpha_{p_1-1}, \alpha_{l_1}^{'}, \alpha_{l_1+1}^{'}, \ldots, \alpha_{l_2-1}^{'}, \underbrace{\alpha^{*}, \alpha^{*}, \ldots\alpha^{*}}_{p_2\; \text{times}}\}$. Hence, $\mathfrak{F}$ is CT. □

5.   Conclusions

In this work, we have introduced the notions of MCT, TMCT, MCM, and AMSP to IFSs and studied their interrelations. In Example 3.5, we have given an example of an IFS that is not CT but MCM. In Theorem 3.7, we proved that a TMCT IFS is MCM if the constituent maps are Lipschitz. For an iterated function system $\mathfrak{F}$, we show that $\mathfrak{F}\times\mathfrak{F}$ is MCT if $\mathfrak{F}$ is MCM. We also showed that $\mathfrak{F}\times\mathfrak{F}$ is MCT if $\mathfrak{F}$ is TMCT. Lastly, we prove that an IFS $\mathfrak{F}$, one of whose constituent maps $g_\alpha$ is surjective and has AMSP, is CT.

Author contributions

Thiyam Thadoi Devi: ideas, states, proof, and examples; Khundrakpam Binod Mangang: ideas, conceptualization, states, examples, and submission; Sonika Akoijam: states, proofs, and first draft; Lalhmangaihzuala: states, proofs, and edition; Phinao Ramwungzan: states, proofs, and examples; Jay Prakash Singh: revision and draft of the manuscript. All authors have read and approved the final version of the manuscript.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

We would like to thank the anonymous referees for their careful reading and valuable suggestions that have help us to improve the initial manuscript.

Conflict of interest

The authors declare no conficts of interest.